5. 7a Numerical Integration. Trapezoidal sums 5.7a Numerical Integration. Trapezoidal sums. Trapezoidal Rule (optional). Rita Korsunsky Rita Korsunsky

Trapezoidal approximation With the Trapezoidal approximation, instead of approximating area by using rectangles (as you do with the left, right, and midpoint Riemann sum methods), you approximate area with trapezoids.

Example 1 3 5 7 10 11 20 22

Example 2 (seconds) 20 24 25 28 35 40 (meters/ seconds) 4 6 3 8 12 12 10 8 6 4 2 …20 24 28 32 36 40

Example 3

Example 4

Trapezoidal Rule (not tested on AP Exam) If all subintervals are of equal length, then we use this formula: Trapezoidal Rule Conditions: Let f be continuous on [a,b]. If a regular partition of [a,b] is determined by a = ao, a1, …, an = b f (x) a b a0 a2 a3 a4 a5 a6 a1 x Trapezoidal Rule gives the sum of the areas of the trapezoids under the curve. Rita Korsunsky

Proof of Trapezoidal Rule 1. Area of each individual trapezoid: f(x) x From the diagram, b1 b2 a0 h a1 Rita Korsunsky

Trapezoidal Rule 2.Since we divide the curve equally in the x-direction, and since there are in total of n blocks. a1-a0=…=an - an-1 will always be (b-a)/n. Let’s find and add the areas of 3 trapezoids: f(x) x a0 = a a1 a2 a3 = b Rita Korsunsky

Trapezoidal Rule Yielding the form… link Yielding the general form: Rita Korsunsky

Applications of the Trapezoidal Rule Yielding the general form: Applications of the Trapezoidal Rule The values of the function are given in the table below. x 1 2 3 4 5 6 7 8 9 10 f(x) 20 19.5 18 15.5 12 7.5 -4.5 -12 -20.5 -30 Rita Korsunsky